Why did Severus Snape killed Albus Dumbledoor?

I believe Dumbledore did ask Snape to kill him.

yes, because dumbledore wants snape to have the elder wand and kill Voldemort, but elder wand knows dumbledore did it in purpose, so snape can get the elder wand. elder wand knows so it does not work. because they did in purpose. Draco Malfoy use his wand and did the expelliarmus, so dumbledore has no wand in his hand. and then snape killed dumbledore, but it will not work for dumbledore because he did in purpose, so it will only work for Draco Malfoy.

We learn in the next book, Harry Potter and the Deathly Hallows, that Dumbledore asked Snape to kill him, because Dumbledore was already dying, but also because Snape had to prove to the Deatheaters and Voldemort that he is truly on their side and that he isn't a spy for Dumbledore. But we must also remember that Narcissa Malfoy made an Unbreakable Vow with Snape that he must help Draco with his task. His task was to kill Dumbledore. So when Draco couldn't kill Dumbledore, Snape had to do it. An Unbreakable Vow kills you if you don't follow the vow that you made.

Dumbledore knew he was dying because of his hand which he hurt from trying to destroy the ring horcrux. Since he knew he was dying, he asked Snape to kill him of necessary, in this case, when Draco couldn't do it, Snape did.